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Theorem brcogw 4808
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )

Proof of Theorem brcogw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 1001 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A  e.  V )
2 simpl2 1002 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  B  e.  W )
3 breq2 4019 . . . . . 6  |-  ( x  =  X  ->  ( A D x  <->  A D X ) )
4 breq1 4018 . . . . . 6  |-  ( x  =  X  ->  (
x C B  <->  X C B ) )
53, 4anbi12d 473 . . . . 5  |-  ( x  =  X  ->  (
( A D x  /\  x C B )  <->  ( A D X  /\  X C B ) ) )
65spcegv 2837 . . . 4  |-  ( X  e.  Z  ->  (
( A D X  /\  X C B )  ->  E. x
( A D x  /\  x C B ) ) )
76imp 124 . . 3  |-  ( ( X  e.  Z  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
873ad2antl3 1162 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  E. x ( A D x  /\  x C B ) )
9 brcog 4806 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
109biimpar 297 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  E. x
( A D x  /\  x C B ) )  ->  A
( C  o.  D
) B )
111, 2, 8, 10syl21anc 1247 1  |-  ( ( ( A  e.  V  /\  B  e.  W  /\  X  e.  Z
)  /\  ( A D X  /\  X C B ) )  ->  A ( C  o.  D ) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2158   class class class wbr 4015    o. ccom 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-co 4647
This theorem is referenced by: (None)
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